PL EN


Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników
Czasopismo
2008 | 6 | 2 | 272-280
Tytuł artykułu

Norm conditions for uniform algebra isomorphisms

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
In recent years much work has been done analyzing maps, not assumed to be linear, between uniform algebras that preserve the norm, spectrum, or subsets of the spectra of algebra elements, and it is shown that such maps must be linear and/or multiplicative. Letting A and B be uniform algebras on compact Hausdorff spaces X and Y, respectively, it is shown here that if λ ∈ ℂ / {0} and T: A → B is a surjective map, not assumed to be linear, satisfying $$ \left\| {T(f)T(g) + \lambda } \right\| = \left\| {fg + \lambda } \right\|\forall f,g \in A, $$ then T is an ℝ-linear isometry and there exist an idempotent e ∈ B, a function κ ∈ B with κ 2 = 1, and an isometric algebra isomorphism $$ \tilde T:{\rm A} \to Be \oplus \bar B(1 - e) $$ such that $$ T(f) = \kappa \left( {\tilde T(f)e + \gamma \overline {\tilde T(f)} (1 - e)} \right) $$ for all f ∈ A, where γ = λ / |λ|. Moreover, if T is unital, i.e. T(1) = 1, then T(i) = i implies that T is an isometric algebra isomorphism whereas T(i) = −i implies that T is a conjugate-isomorphism.
Wydawca
Czasopismo
Rocznik
Tom
6
Numer
2
Strony
272-280
Opis fizyczny
Daty
wydano
2008-06-01
online
2008-04-15
Twórcy
Bibliografia
  • [1] Browder A., Introduction to function algebras, W.A. Benjamin Inc., New York-Amsterdam, 1969
  • [2] Hatori O., Miura T., Takagi H., Characterization of isometric isomorphisms between uniform algebras via non-linear range preserving properties, Proc. Amer. Math. Soc., 2006, 134, 2923–2930 http://dx.doi.org/10.1090/S0002-9939-06-08500-5
  • [3] Hatori O., Miura T., Takagi H., Unital and multiplicatively spectrum-preserving surjections between semi-simple commutative Banach algebras are linear and multiplicative, J. Math. Anal. Appl., 2007, 326, 281–296 http://dx.doi.org/10.1016/j.jmaa.2006.02.084
  • [4] Hatori O., Miura T., Takagi H., Multiplicatively spectrum-preserving and norm-preserving maps between invertible groups of commutative Banach algebras, preprint
  • [5] Hatori O., Miura T., Takagi H., Polynomially spectrum-preserving maps between commutative Banach algebras, preprint
  • [6] Honma D., Surjections on the algebras of continuous functions which preserve peripheral spectrum, Contemp. Math., 2007, 435, 199–205
  • [7] Honma D., Norm-preserving surjections on algbras of continuous functions, Rocky Mountain J. Math., to appear
  • [8] Jiménez-Vargas A., Luttman A., Villegas-Vallecillos M., Weakly peripherally multiplicative surjections of pointed Lipschitz algebras, preprint
  • [9] Kowalski S., Słodkowski Z., A Characterization of maximal ideals in commutative Banach algebras, Studia Math., 1980, 67, 215–223
  • [10] Lambert S., Luttman A., Tonev T., Weakly peripherally-multiplicative operators between uniform algebras, Contemp. Math., 2007, 435, 265–281
  • [11] Li C-K., Tsing N-K., Linear preserver problems: a brief introduction and some special techniques, Linear Algebra Appl., 1992, 162/164, 217–235 http://dx.doi.org/10.1016/0024-3795(92)90377-M
  • [12] Luttman A., Tonev T., Uniform algebra isomorphisms and peripheral multiplicativity, Proc. Amer. Math. Soc., 2007, 135, 3589–3598 http://dx.doi.org/10.1090/S0002-9939-07-08881-8
  • [13] Mazur S., Ulam S., Sur les transformations isométriques d’espaces vectoriels normés, C. R. Math. Acad. Sci. Paris, 1932, 194, 946–948
  • [14] Molnár L., Some characterizations of the automorphisms of B(H) and C(X), Proc. Amer. Math. Soc., 2001, 130, 111–120 http://dx.doi.org/10.1090/S0002-9939-01-06172-X
  • [15] Rao N.V., Roy A.K., Multiplicatively spectrum-preserving maps of function algebras, Proc. Amer. Math. Soc., 2005, 133, 1135–1142 http://dx.doi.org/10.1090/S0002-9939-04-07615-4
  • [16] Rao N.V., Roy A.K., Multiplicatively spectrum-preserving maps of function algebras II, Proc. Edinb. Math. Soc., 2005, 48, 219–229 http://dx.doi.org/10.1017/S0013091504000719
  • [17] Rao N.V., Tonev T., Toneva E., Uniform algebra isomorphisms and peripheral spectra, Contemp. Math., 2007, 427, 401–416
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_2478_s11533-008-0016-x
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.